On the exact and approximate distributions of the product of a Wishart matrix with a normal vector

In this paper we consider the distribution of the product of a Wishart random matrix and a Gaussian random vector. We derive a stochastic representation for the elements of the product. Using this result, the exact joint density for an arbitrary linear combination of the elements of the product is obtained. Furthermore, the derived stochastic representation allows us to simulate samples of arbitrary size by generating independently distributed chi-squared random variables and standard multivariate normal random vectors for each element of the sample. Additionally to the Monte Carlo approach, we suggest another approximation of the density function, which is based on the Gaussian integral and the third order Taylor expansion. We investigate, with a numerical study, the properties of the suggested approximations. A good performance is documented for both methods.